scholarly journals The Global and Exponential Attractors for the Higher-order Kirchhoff-type Equation with Strong Linear Damping

2017 ◽  
Vol 9 (4) ◽  
pp. 145 ◽  
Author(s):  
Guoguang Lin ◽  
Yunlong Gao
Keyword(s):  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Attractors ◽  
Higher Order ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Kirchhoff Type ◽  
Uniqueness Of The Solution ◽  
Initial Boundary Value

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: ${u_{tt}} + {( - \Delta )^m}{u_t} + {\left( {\alpha + \beta\left\| {{\nabla ^m}u} \right\|^2} \right)^{q}}{( - \Delta )^m}u + g(u) = f(x)$. At first, we do priori estimation for the equations to obtain two lemmas and prove the existence and uniqueness of the solution by the lemmas and the Galerkin method. Then, we obtain to the existence of the global attractor in $H_0^m(\Omega ) \times {L^2}(\Omega )$ according to some of the attractor theorem. In this case, we consider that the estimation of the upper bounds of Hausdorff  for the global attractors are obtained. At last, we also establish the existence of a fractal exponential attractor with the non-supercritical and critical cases.

Reconstruction of a Space-Dependent Coefficient in a Linear Benjamin–Bona–Mahony Type Equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Felipe Alexander Pipicano ◽  
Juan Carlos Muñoz Grajales ◽  
Anibal Sosa
Keyword(s):  
Constrained Optimization ◽  
Regularization Parameter ◽  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Time Stepping ◽  
Uniqueness Of The Solution ◽  
Initial Boundary Value ◽  
Bbm Equation

Abstract In this paper, we consider the problem of reconstructing a space-dependent coefficient in a linear Benjamin–Bona–Mahony (BBM)-type equation from a single measurement of its solution at a given time. We analyze the well-posedness of the forward initial-boundary value problem and characterize the inverse problem as a constrained optimization one. Our objective consists on reconstructing the variable coefficient in the BBM equation by minimizing an appropriate regularized Tikhonov-type functional constrained by the BBM equation. The well-posedness of the forward problem is studied and approximated numerically by combining a finite-element strategy for spatial discretization using the Python-FEniCS package, together with a second-order implicit scheme for time stepping. The minimization process of the Tikhonov-regularization adopted is performed by using an iterative L-BFGS-B quasi-Newton algorithm as described for instance by Byrd et al. (1995) and Zhu et al. (1997). Numerical simulations are presented to demonstrate the robustness of the proposed method with noisy data. The local stability and uniqueness of the solution to the constrained optimization problem for a fixed value of the regularization parameter are also proved and illustrated numerically.

scholarly journals The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation with nonlinear strongly damped terms

2016 ◽  
Vol 12 (10) ◽  
pp. 6658-6673
Author(s):  
Wei Wang ◽  
Ling Chen ◽  
Guoguang Lin
Keyword(s):  
Fractal Dimensions ◽  
Type Equation ◽  
Initial Boundary ◽  
Global Attractors ◽  
Time Behavior ◽  
Initial Boundary Value Problems ◽  
Long Time ◽  
Uniqueness Of The Solution ◽  
Hausdorff And Fractal Dimensions ◽  
Initial Boundary Value

In this paper ,we study the long time behavior of solution to the initial boundary value problems for higher -orderkirchhoff-type equation with nonlinear strongly dissipation:At first ,we prove the existence and uniqueness of the solution by priori estimate and Galerkin methodthen we establish the existence of global attractors ,at last,we consider that estimation of upper bounds of Hausdorff and fractal dimensions for the global attractors are obtain.

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

scholarly journals Homotopy Analysis Method for a Fractional Order Equation with Dirichlet and Non-Local Integral Conditions

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1167
Author(s):  
Said Mesloub ◽  
Saleem Obaidat
Keyword(s):  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Uniqueness Of The Solution ◽  
Non Local ◽  
Initial Boundary Value

The main purpose of this paper is to obtain some numerical results via the homotopy analysis method for an initial-boundary value problem for a fractional order diffusion equation with a non-local constraint of integral type. Some examples are provided to illustrate the efficiency of the homotopy analysis method (HAM) in solving non-local time-fractional order initial-boundary value problems. We also give some improvements for the proof of the existence and uniqueness of the solution in a fractional Sobolev space.

scholarly journals Evolutionary equation of inertial waves in 3-D multiply connected domain with Dirichlet boundary condition

2001 ◽  
Vol 25 (9) ◽  
pp. 587-602
Author(s):  
Pavel A. Krutitskii
Keyword(s):  
Connected Domain ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Evolutionary Equation ◽  
Inertial Waves ◽  
Rotating Fluids ◽  
Multiply Connected Domain ◽  
Multiply Connected ◽  
Uniqueness Of The Solution ◽  
Initial Boundary Value

We study initial-boundary value problem for an equation of composite type in 3-D multiply connected domain. This equation governs nonsteady inertial waves in rotating fluids. The solution of the problem is obtained in the form of dynamic potentials, which density obeys the uniquely solvable integral equation. Thereby the existence theorem is proved. Besides, the uniqueness of the solution is studied. All results hold for interior domains and for exterior domains with appropriate conditions at infinity.

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